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# A general LIL for trimmed sums of random fields in banach spaces

Acta Mathematica Hungarica
Authors: K.-A. Fu and L.-X. Zhang

## Abstract

Given a field of independent identically distributed (i.i.d.) random variables
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\}$$ \end{document}
indexed by d-tuples of positive integers and taking values in a separable Banach space B, let
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$X_{\bar n}^{(r)} = X_{\bar m}$$ \end{document}
is the r-th maximum of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\}$$ \end{document}
and let
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right)$$ \end{document}
be the trimmed sums, where
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} }$$ \end{document}
. This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.
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# LIL behavior for weakly dependent random variables in Banach spaces

Acta Mathematica Hungarica
Author: K. Fu

## Abstract

Given a sequence of identically distributed ψ-mixing random variables {X n; n ≧ 1} with values in a type 2 Banach space B, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.

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# A nonclassical LIL for sums of B-valued random variables when extreme terms are excluded

Acta Mathematica Hungarica
Author: Ke-Ang Fu

## Abstract

Let {X,X n; n≧1} be a sequence of B-valued i.i.d. random variables. Denote if ∥X m∥ is the r-th maximum of {∥X k∥; kn}, and let be the trimmed sums, where . Given a sequence of positive constants {h(n), n≧1}, which is monotonically approaching infinity and not asymptotically equivalent to loglogn, a limit result for is derived.

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# Strong approximation for sums of asymptotically negatively dependent Gaussian sequences with applications

Acta Mathematica Hungarica
Authors: K.-A. Fu and L.-X. Zhang

## Abstract

By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences are derived.

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# A general LIL for B-valued geometrically weighted series under dependent assumption

Acta Mathematica Hungarica
Author: Ke-Ang Fu

## Abstract

Let {X,X n; n≧0} be a sequence of identically distributed ψ-mixing dependent random variables taking values in a type 2 Banach space B with topological dual B . Considering the geometrically weighted series for 0<β<1, a general law of the iterated logarithm for ξ(β) is obtained without second moment.

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# On almost sure limiting behavior of weighted sums of random fields

Acta Mathematica Hungarica
Authors: Z. Lagodowski and P. Matuła

## Abstract

We study necessary and sufficient conditions for the almost sure convergence of averages of independent random variables with multidimensional indices obtained by certain summability methods.

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# Joint asymptotic behavior of local and occupation times of random walk in higher dimension

Studia Scientiarum Mathematicarum Hungarica
Authors: Endre Csáki, Antónia Földes, and Pál Révész

Considering a simple symmetric random walk in dimension d ≧ 3, we study the almost sure joint asymptotic behavior of two objects: first the local times of a pair of neighboring points, then the local time of a point and the occupation time of the surface of the unit ball around it.

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# Merging asymptotic expansions for cooperative gamblers in generalized St. Petersburg games

Acta Mathematica Hungarica
Authors: S. Csörgő and P. Kevei

## Abstract

Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any one of n cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations even for very small n.

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# On the rate of convergence of the St. Petersburg game

Periodica Mathematica Hungarica
Authors: László Györfi and Péter Kevei

## Abstract

We investigate the repeated and sequential portfolio St. Petersburg games. For the repeated St. Petersburg game, we show an upper bound on the tail distribution, which implies a strong law for a truncation. Moreover, we consider the problem of limit distribution. For the sequential portfolio St. Petersburg game, we obtain tight asymptotic results for the growth rate of the game.

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# Random walk and Brownian local times in Wiener sheets: A tribute to my almost surely most visited 75 years young best friends, Endre Csáki and Pál Révész

Periodica Mathematica Hungarica
Author: Miklós Csörgő
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